Quaternionic contact hypersurfaces in hyper-K\"ahler manifolds

We describe explicitly all quaternionic contact hypersurfaces (qc-hypersurfaces) in the flat quaternion space $\Hnn$ and the quaternion projective space. We show that up to a quaternionic affine transformation a qc-hypersurface in $\Hnn$ is contained in one of the three qc-hyperquadrics in $\Hnn$. Moreover, we show that an embedded qc-hypersurface in a hyper-K\"ahler manifold is qc-conformal to a qc-Einstein space and the Riemannian curvature tensor of the ambient hyper-K\"ahler metric is degenerate along the hypersurface.